Python scipy.linalg.orthogonal_procrustes() Examples

def test_orthogonal_procrustes_exact_example():
    # Check a small application.
    # It uses translation, scaling, reflection, and rotation.
    #
    #         |
    #   a  b  |
    #         |
    #   d  c  |        w
    #         |
    # --------+--- x ----- z ---
    #         |
    #         |        y
    #         |
    #
    A_orig = np.array([[-3, 3], [-2, 3], [-2, 2], [-3, 2]], dtype=float)
    B_orig = np.array([[3, 2], [1, 0], [3, -2], [5, 0]], dtype=float)
    A, A_mu = _centered(A_orig)
    B, B_mu = _centered(B_orig)
    R, s = orthogonal_procrustes(A, B)
    scale = s / np.square(norm(A))
    B_approx = scale * np.dot(A, R) + B_mu
    assert_allclose(B_approx, B_orig, atol=1e-8) 

def test_orthogonal_procrustes_stretched_example():
    # Try again with a target with a stretched y axis.
    A_orig = np.array([[-3, 3], [-2, 3], [-2, 2], [-3, 2]], dtype=float)
    B_orig = np.array([[3, 40], [1, 0], [3, -40], [5, 0]], dtype=float)
    A, A_mu = _centered(A_orig)
    B, B_mu = _centered(B_orig)
    R, s = orthogonal_procrustes(A, B)
    scale = s / np.square(norm(A))
    B_approx = scale * np.dot(A, R) + B_mu
    expected = np.array([[3, 21], [-18, 0], [3, -21], [24, 0]], dtype=float)
    assert_allclose(B_approx, expected, atol=1e-8)
    # Check disparity symmetry.
    expected_disparity = 0.4501246882793018
    AB_disparity = np.square(norm(B_approx - B_orig) / norm(B))
    assert_allclose(AB_disparity, expected_disparity)
    R, s = orthogonal_procrustes(B, A)
    scale = s / np.square(norm(B))
    A_approx = scale * np.dot(B, R) + A_mu
    BA_disparity = np.square(norm(A_approx - A_orig) / norm(A))
    assert_allclose(BA_disparity, expected_disparity) 

def _difference_norm(self, X1, X2):
        if self.embedding in ["ase"]:
            if self.test_case == "rotation":
                R = orthogonal_procrustes(X1, X2)[0]
                return np.linalg.norm(X1 @ R - X2)
            elif self.test_case == "scalar-rotation":
                R, s = orthogonal_procrustes(X1, X2)
                return np.linalg.norm(s / np.sum(X1 ** 2) * X1 @ R - X2)
            elif self.test_case == "diagonal-rotation":
                normX1 = np.sum(X1 ** 2, axis=1)
                normX2 = np.sum(X2 ** 2, axis=1)
                normX1[normX1 <= 1e-15] = 1
                normX2[normX2 <= 1e-15] = 1
                X1 = X1 / np.sqrt(normX1[:, None])
                X2 = X2 / np.sqrt(normX2[:, None])
                R = orthogonal_procrustes(X1, X2)[0]
                return np.linalg.norm(X1 @ R - X2)
        else:
            # in the omni case we don't need to align
            return np.linalg.norm(X1 - X2) 

def test_orthogonal_procrustes_ndim_too_large():
    np.random.seed(1234)
    A = np.random.randn(3, 4, 5)
    B = np.random.randn(3, 4, 5)
    assert_raises(ValueError, orthogonal_procrustes, A, B) 

def test_orthogonal_procrustes_ndim_too_small():
    np.random.seed(1234)
    A = np.random.randn(3)
    B = np.random.randn(3)
    assert_raises(ValueError, orthogonal_procrustes, A, B) 

def test_orthogonal_procrustes_shape_mismatch():
    np.random.seed(1234)
    shapes = ((3, 3), (3, 4), (4, 3), (4, 4))
    for a, b in permutations(shapes, 2):
        A = np.random.randn(*a)
        B = np.random.randn(*b)
        assert_raises(ValueError, orthogonal_procrustes, A, B) 

def test_orthogonal_procrustes_checkfinite_exception():
    np.random.seed(1234)
    m, n = 2, 3
    A_good = np.random.randn(m, n)
    B_good = np.random.randn(m, n)
    for bad_value in np.inf, -np.inf, np.nan:
        A_bad = A_good.copy()
        A_bad[1, 2] = bad_value
        B_bad = B_good.copy()
        B_bad[1, 2] = bad_value
        for A, B in ((A_good, B_bad), (A_bad, B_good), (A_bad, B_bad)):
            assert_raises(ValueError, orthogonal_procrustes, A, B) 

def test_orthogonal_procrustes_scale_invariance():
    np.random.seed(1234)
    m, n = 4, 3
    for i in range(3):
        A_orig = np.random.randn(m, n)
        B_orig = np.random.randn(m, n)
        R_orig, s = orthogonal_procrustes(A_orig, B_orig)
        for A_scale in np.square(np.random.randn(3)):
            for B_scale in np.square(np.random.randn(3)):
                R, s = orthogonal_procrustes(A_orig * A_scale, B_orig * B_scale)
                assert_allclose(R, R_orig) 

def test_orthogonal_procrustes():
    np.random.seed(1234)
    for m, n in ((6, 4), (4, 4), (4, 6)):
        # Sample a random target matrix.
        B = np.random.randn(m, n)
        # Sample a random orthogonal matrix
        # by computing eigh of a sampled symmetric matrix.
        X = np.random.randn(n, n)
        w, V = eigh(X.T + X)
        assert_allclose(inv(V), V.T)
        # Compute a matrix with a known orthogonal transformation that gives B.
        A = np.dot(B, V.T)
        # Check that an orthogonal transformation from A to B can be recovered.
        R, s = orthogonal_procrustes(A, B)
        assert_allclose(inv(R), R.T)
        assert_allclose(A.dot(R), B)
        # Create a perturbed input matrix.
        A_perturbed = A + 1e-2 * np.random.randn(m, n)
        # Check that the orthogonal procrustes function can find an orthogonal
        # transformation that is better than the orthogonal transformation
        # computed from the original input matrix.
        R_prime, s = orthogonal_procrustes(A_perturbed, B)
        assert_allclose(inv(R_prime), R_prime.T)
        # Compute the naive and optimal transformations of the perturbed input.
        naive_approx = A_perturbed.dot(R)
        optim_approx = A_perturbed.dot(R_prime)
        # Compute the Frobenius norm errors of the matrix approximations.
        naive_approx_error = norm(naive_approx - B, ord='fro')
        optim_approx_error = norm(optim_approx - B, ord='fro')
        # Check that the orthogonal Procrustes approximation is better.
        assert_array_less(optim_approx_error, naive_approx_error) 

def test_orthogonal_procrustes_skbio_example():
    # This transformation is also exact.
    # It uses translation, scaling, and reflection.
    #
    #   |
    #   | a
    #   | b
    #   | c d
    # --+---------
    #   |
    #   |       w
    #   |
    #   |       x
    #   |
    #   |   z   y
    #   |
    #
    A_orig = np.array([[4, -2], [4, -4], [4, -6], [2, -6]], dtype=float)
    B_orig = np.array([[1, 3], [1, 2], [1, 1], [2, 1]], dtype=float)
    B_standardized = np.array([
        [-0.13363062, 0.6681531],
        [-0.13363062, 0.13363062],
        [-0.13363062, -0.40089186],
        [0.40089186, -0.40089186]])
    A, A_mu = _centered(A_orig)
    B, B_mu = _centered(B_orig)
    R, s = orthogonal_procrustes(A, B)
    scale = s / np.square(norm(A))
    B_approx = scale * np.dot(A, R) + B_mu
    assert_allclose(B_approx, B_orig)
    assert_allclose(B / norm(B), B_standardized)